UPSEE Solved Model Paper 1

Let radius of cylinder = r
Let height of cylinder = 2h
Then from geometry
r2+h2 = R2 ⇒ r2 = R2−h2
Volume of cylinder V = πr2(2h)
V = π(2h)(R2–h2)
this volume will be maximum, when

dV

dh

= 0

dV

dh

= 2πR2−6πh2
0 = 2πR2−6πh2
R2 = 3h2
h =

R

√3

r2 = R2−(R∕√3)2 =

2

3

R2
r = √

2

3

R
So, radius ‘r’ and height ‘2h’ of the cylinder having maximum possible volume is
√

2

3

R and

2

√3

R respectively.
Now volume of cylinder
V = πr2(2h)
= π(

2

3

R2).2

R

√3

=

4πR3

3√3

Mass of cylinder
M' = (

M

4 3 πR3

)×

4

3√3

πR3
=

M

√3

Moment of inertia of solid cylinder about its axis
I =

M′r2

2

=

1

2

(M∕√3)(

2

3

R2)
1 =

MR2

3√3